The Time-Dependent Symbol of a Non-Homogeneous It\^o Process and corresponding Maximal Inequalities

TL;DR

This paper introduces a time-dependent probabilistic symbol for non-homogeneous Itô processes and derives maximal inequalities, extending classical results like Blumenthal-Getoor indices to non-homogeneous settings.

Contribution

It establishes the existence of a time-dependent symbol for non-homogeneous Itô processes and derives maximal inequalities applicable to path properties.

Findings

Generalized Blumenthal-Getoor indices for non-homogeneous processes

Analyzed asymptotic behavior of sample paths

Proved finiteness of p-variation and existence of exponential moments

Abstract

The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article we show the existence of such a time-dependent symbol for non-homogeneous It\^o processes. Moreover, for this class of processes we derive maximal inequalities which we apply to generalize the Blumenthal-Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample patsh, the existence of exponential moments and the finiteness of p-variation. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.